import numpy as np
import matplotlib.pyplot as plt
%matplotlib notebook

Configurable parameters for pure pursuit

• How fast do you want the robot to move? It is fixed at $v_{max}$ in this exercise
• When can we declare the goal has been reached?
• What is the lookahead distance? Determines the next position on the reference path that we want the vehicle to catch up to

vmax = 0.75
goal_threshold = 0.05
lookahead = 3.0

#You know what to do!
def simulate_unicycle(pose, v,w, dt=0.1):
    x, y, t = pose
    return x + np.cos(t)*dt, y + np.sin(t)*dt, t+w*dt

class PurePursuitTracker(object):
    def __init__(self, x, y, v, lookahead = 3.0):
        """
        Tracks the path defined by x, y at velocity v
        x and y must be numpy arrays
        v and lookahead are floats
        """
        self.length = len(x)
        self.ref_idx = 0 #index on the path that tracker is to track
        self.lookahead = lookahead
        self.x, self.y = x, y
        self.v, self.w = v, 0

    def update(self, xc, yc, theta):
        """
        Input: xc, yc, theta - current pose of the robot
        Update v, w based on current pose
        Returns True if trajectory is over.
        """
        #Calculate ref_x, ref_y using current ref_idx
        #Check if we reached the end of path, then return TRUE
        #Two conditions must satisfy
        #1. ref_idx exceeds length of traj
        #2. ref_x, ref_y must be within goal_threshold
        # Write your code to check end condition
        ref_x, ref_y = self.x[self.ref_idx], self.y[self.ref_idx]
        goal_x, goal_y = self.x[-1], self.y[-1]

        if (self.ref_idx > self.length) or \
            (np.linalg.norm([ref_x-goal_x, ref_y-goal_y])) < goal_threshold:
              return True
        
        #End of path has not been reached
        #update ref_idx using np.hypot([ref_x-xc, ref_y-yc]) < lookahead
        if np.hypot(ref_x-xc, ref_y-yc) < lookahead:
          self.ref_idx = self.ref_idx + 1

        #Find the anchor point
        # this is the line we drew between (0, 0) and (x, y)
        anchor = np.asarray([ref_x - xc, ref_y - yc])
        #Remember right now this is drawn from current robot pose
        #we have to rotate the anchor to (0, 0, pi/2)
        #code is given below for this
        theta = np.pi/2 - theta
        rot = np.asarray([[np.cos(theta), -np.sin(theta)], [np.sin(theta), np.cos(theta)]])
        anchor = np.dot(rot, anchor)
        L = np.sqrt(anchor[0] ** 2 + anchor[1] **2) # dist to reference path
        X = anchor[0] #cross-track error
        #from the derivation in notes, plug in the formula for omega
        self.w = -1*2*self.v*X/L**2
        
        return False

Visualize given trajectory

x = np.arange(0, 50, 0.5)
y = [np.sin(idx / 5.0) * idx / 2.0 for idx in x]

#write code here
target_speed = 10.0 / 3.6  # [m/s]

Run the tracker simulation

1. Instantiate the tracker class
2. Initialize some starting pose
3. Simulate robot motion 1 step at a time - get $v$, $\omega$ from tracker, predict new pose using $v$, $\omega$, current pose in simulate_unicycle()
4. Stop simulation if tracker declares that end-of-path is reached
5. Record all parameters

#write code to instantiate the tracker class
tracker = PurePursuitTracker(x,y,target_speed) 
pose = -1, 0, np.pi/2 #arbitrary initial pose
x0,y0,t0 = pose # record it for plotting
traj =[]
while True:
    #write the usual code to obtain successive poses
    pose = simulate_unicycle(pose,tracker.v,tracker.w)
    if tracker.update(*pose):
        print("ARRIVED!!")
        break
    traj.append([*pose, tracker.w, tracker.ref_idx])
print(traj)

ARRIVED!!
[[-1.0, 0.1, 1.5707963267948966, -5.500550055005501, 1], [-1.0, 0.2, 1.0207413212943464, -3.337864590197384, 2], [-0.947726587832202, 0.28524957700971604, 0.686954862274608, -2.001179836401396, 3], [-0.8704085098760229, 0.3486681454353926, 0.48683687863446834, -1.204012471451654, 4], [-0.782026800984548, 0.3954514066879237, 0.3664356314893029, -0.7195251865461889, 5], [-0.6886657658784027, 0.4312804050538995, 0.29448311283468404, -0.41668825588076797, 5], [-0.5929705364588261, 0.46030493030987246, 0.25281428724660726, -0.3703893187219352, 5], [-0.4961493044759516, 0.48531790768799343, 0.21577535537441372, -0.32763247585479316, 6], [-0.39846823642785745, 0.5067283944873072, 0.1830121077889344, -0.13522546170430105, 6], [-0.3001382390149465, 0.5249276144919376, 0.1694895616185043, -0.12088902527402193, 6], [-0.20157113944749616, 0.5417955391674965, 0.1574006590911021, -0.10766910619117476, 6], [-0.10280733243684964, 0.5574706922692614, 0.14663374847198463, -0.0955145228101153, 6], [-0.003880480323900898, 0.5720815762610933, 0.1370822961909731, -0.0843737983916716, 6], [0.09518141230239, 0.5857469130405956, 0.12864491635180594, -0.07419550460680277, 7], [0.19435507714133748, 0.5985759505108079, 0.12122536589112566, 0.06205338903729048, 7], [0.2936211970686692, 0.6106688176056123, 0.12743070479485472, 0.05532260833404519, 7], [0.39281036596301067, 0.6233774278384349, 0.13296296562825924, 0.04913217853726232, 7], [0.49192770998583707, 0.6366345811475235, 0.13787618348198546, 0.04345517124878324, 7], [0.5909787226588453, 0.6503785575900073, 0.1422217006068638, 0.038265037543651986, 8], [0.6899690756204551, 0.6645528307825175, 0.14604820436122898, 0.13480816626645153, 8], [0.7889044660908653, 0.6791057862372515, 0.15952902098787414, 0.1204421891690358, 8], [0.8876346870319087, 0.6949911088003202, 0.1715732399047777, 0.10720115890138485, 8], [0.9861664253213516, 0.7120643788345211, 0.1822933557949162, 0.09503314995825694, 8], [1.0845094780599491, 0.7301929192626256, 0.1917966707907419, 0.08388596188348175, 9], [1.1826758113671998, 0.7492552120158532, 0.20018526697909006, 0.15392427807242437, 9], [1.2806787867826865, 0.769140302151719, 0.21557769478633249, 0.1378769229305384, 9], [1.3783640849319796, 0.790531480835553, 0.22936538707938633, 0.12305425867896316, 9], [1.4757451725733446, 0.8132674384750458, 0.24167081294728265, 0.10940262392902858, 9], [1.572839118834347, 0.8371999602699927, 0.2526110753401855, 0.09686755552079045, 9], [1.6696654317528845, 0.8621932619046593, 0.26229783089226455, 0.08539426734078492, 10], [1.7662451017267888, 0.8881233091034537, 0.27083725762634303, 0.14248348037356445, 10], [1.8625998252928477, 0.9148771358672715, 0.28508560566369945, 0.12755873952662017, 10], [1.9585635833399326, 0.9430010961410404, 0.2978414796163615, 0.11378057342921587, 10], [2.054160798286353, 0.9723468367942916, 0.30921953695928306, 0.10109766512882776, 10], [2.1494179349672153, 1.0027783650399043, 0.3193293034721658, 0.08945821210174397, 10], [2.244362553293966, 1.0341713491412547, 0.3282751246823402, 0.0788103133303894, 11], [2.3390225402682616, 1.0664124233483303, 0.3361561560153792, 0.12327435530227795, 11], [2.43342549727515, 1.0993985069042107, 0.348483591545607, 0.11033608469826331, 11], [2.5274146578492784, 1.1335458010074146, 0.3595172000154333, 0.09839629471785417, 11], [2.6210213371081683, 1.1687280351337936, 0.3693568294872187, 0.08740918839693626, 11], [2.714277310421736, 1.2048296063092867, 0.37809774832691234, 0.07732887865106326, 11], [2.807214164323735, 1.2417449308615436, 0.3858306361920187, 0.06810964389319027, 12], [2.8998627803209907, 1.2793778148035573, 0.39264160058133774, 0.10032979100803108, 12], [2.992252933126619, 1.3176408474140624, 0.4026745796821408, 0.0897920918837374, 12], [3.0842545501778567, 1.3568288871705676, 0.41165378887051457, 0.08007021969071478, 12], [3.175900585505484, 1.3968414378106768, 0.41966081083958606, 0.07112591063282818, 12], [3.2672233050765103, 1.4375865097844955, 0.4267734019028689, 0.06292109651443019, 12], [3.3582539141056253, 1.4789800868199297, 0.4330655115543119, 0.05541804659446, 13], [3.449022269953481, 1.520945615253121, 0.43860731621375787, 0.07583689231189193, 13], [3.539556668413633, 1.563413517196335, 0.44619100544494705, 0.06786301326718565, 13], [3.629766403180059, 1.6065667760926345, 0.45297730677166564, 0.060508437524729765, 13], [3.7196812119297666, 1.6503312270489023, 0.45902815052413865, 0.05374356680390245, 13], [3.809329564433343, 1.6946389339788992, 0.4644025072045289, 0.04753912297395885, 13], [3.898738497977131, 1.7394278009277577, 0.46915641950192477, 0.041866213057758256, 14], [3.987933499672703, 1.7846412023981941, 0.4733430408077006, 0.05081988677609113, 14], [4.076938428836507, 1.8302276322271063, 0.47842502948530974, 0.04545988570491289, 14], [4.1657105399319985, 1.8762657934811113, 0.482971018055801, 0.040518135544524075, 14], [4.254272445512303, 1.9227070346357715, 0.4870228316102534, 0.03597432429151293, 14], [4.342645453389127, 1.9695067299211335, 0.4906202640394047, 0.03180844283891591, 14], [4.4308495310476825, 2.016624037615903, 0.49380110832329627, 0.028000807790888035, 15], [4.518903289926189, 2.0640216699129477, 0.49660118910238504, 0.025527417373847897, 15], [4.606823986588528, 2.1116656737162494, 0.49915393083976983, 0.02281717626107402, 15], [4.694622774078883, 2.1595339608719804, 0.5014356484658772, 0.02032005977338409, 15], [4.7823121111989995, 2.2076024552881197, 0.5034676544432156, 0.018025565258985528, 15], [4.869903591881668, 2.2558490356002805, 0.5052702109691142, 0.015923373184693838, 15], [4.957407963121717, 2.3042534260404417, 0.5068625482875836, 0.014003352254280265, 16], [5.044835147341751, 2.3527970915319996, 0.5082628835130116, -0.000325285334102487, 16], [5.13219426845958, 2.401463136753699, 0.5082303549796013, -0.0002903517950092079, 16], [5.219554972566269, 2.4501263402855624, 0.5082013198001004, -0.00025819880382738997, 16], [5.306917089580983, 2.4987870072631884, 0.5081754999197177, -0.0002286873097929937, 16], [5.394280462979177, 2.547445418545183, 0.5081526311887383, -0.00020168082817144942, 17], [5.481644949110644, 2.5961018319249707, 0.5081324631059212, -0.025851778295426446, 17], [5.569010416530917, 2.644756483320671, 0.5055472852763786, -0.023110940306898947, 17], [5.656501372799326, 2.693185117115559, 0.5032361912456887, -0.02058532024302847, 17], [5.744104018443277, 2.7414114219312955, 0.5011776592213859, -0.01826432884776407, 17], [5.831805753799582, 2.78945729184217, 0.4993512263366095, -0.01613756096517679, 17], [5.919595095383694, 2.8373429003715396, 0.4977374702400918, -0.014194800750377042, 18], [6.007461598316105, 2.8850867760251617, 0.49631799016505407, -0.05380672358480123, 18], [6.095395784183819, 2.9327058788703684, 0.49093731780657396, -0.047966927858281064, 18], [6.183584918689448, 2.979851149631582, 0.48614062502074584, -0.04259644311807287, 18], [6.27199917917064, 3.026572863465343, 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21], [7.354000687233321, 3.5445461244598135, 0.3941042740581606, -0.15145196302875286, 21], [7.446334774905601, 3.5829442527179407, 0.3789590777552853, -0.13481500996197265, 21], [7.539239798081753, 3.6199396128140213, 0.36547757675908804, -0.11952028860339332, 21], [7.63263511647742, 3.6556791497692194, 0.3535255478987487, -0.10550463409987049, 22], [7.7264509139526485, 3.690299897076102, 0.34297508448876163, -0.19119509230994017, 22], [7.820626748202551, 3.7239289357679053, 0.3238555752577676, -0.1706386201835099, 22], [7.915428301266271, 3.755751341973387, 0.3067917132394166, -0.15169342704641237, 22], [8.010759039522393, 3.785951513223613, 0.29162237053477535, -0.13428994734435232, 22], [8.106536908938583, 3.8147021606962377, 0.27819337580034015, -0.11835635710854508, 23], [8.202692222983774, 3.842164053946169, 0.26635774008948565, -0.21979144365503256, 23], [8.299165823656148, 3.868485991057778, 0.2443785957239824, -0.19596475780027708, 23], [8.396194609975582, 3.892681334109234, 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[48.18296906107312, -5.079186515439437, -1.36418215231891, -0.00011509784013650461, 98], [48.20348378799983, -5.177059627064248, -1.3641936621029236, -0.00010463826116265499, 98], [48.2239973884268, -5.2749329748026526, -1.3642041259290398, -9.4879801751647e-05, 98], [48.24450996472296, -5.372806537186447, -1.364213613909215, -8.579308384031225e-05, 98], [48.26502161239577, -5.470680294188753, -1.3642221932175989, -7.734918119040245e-05, 98], [48.285532420378686, -5.568554227163208, -1.364229928135718, -6.951962510103183e-05, 98], [48.30604247131413, -5.666428318784155, -1.364236880098228, -6.227641036304643e-05, 98], [48.32655184183206, -5.764302552987843, -1.3642431077392643, -5.559200148654849e-05, 98], [48.347060602823994, -5.86217691491463, -1.364248666939413, -4.943933920074325e-05, 98], [48.367568819712446, -5.960051390852212, -1.364253610873333, -4.379184724387652e-05, 98], [48.3880765527157, -6.057925968179867, -1.3642579900580574, -3.862343949980926e-05, 99]]

xs,ys,ts,ws,ids = zip(*traj)
plt.figure()
plt.plot(x,y,label='Reference')
plt.quiver(x0,y0, np.cos(t0), np.sin(t0),scale=12)
plt.plot(xs,ys,label='Tracked')
x0,y0,t0 = pose
plt.quiver(x0,y0, np.cos(t0), np.sin(t0),scale=12)
plt.title('Pure Pursuit trajectory')
plt.legend()
plt.grid()

Visualize curvature

plt.figure()
plt.title('Curvature')
plt.plot(np.abs(ws))
plt.grid()

Animate

Make a video to plot the current pose of the robot and reference pose it is trying to track. You can use funcAnimation in matplotlib

# !apt install ffmpeg
from matplotlib import animation
from matplotlib import rc

fig, ax = plt.subplots()
ax.set_xlim(-10, 50)
ax.set_ylim(-22, 22)
line, = ax.plot([], [])

def animation_frame(i):
  line.set_data(xs,ys)
  line.set_data(x,y)
  return line,

anim = animation.FuncAnimation(fig, animation_frame, frames=np.arange(1,len(traj),2))

rc('animation', html = 'jshtml')
rc('animation', embed_limit = 4096)
#HTML(anim.to_jshtml())
fig.show()
anim

Once Loop Reflect

Effect of noise in simulations

What happens if you add a bit of Gaussian noise to the simulate_unicycle() output? Is the tracker still robust?

The noise signifies that $v$, $\omega$ commands did not get realized exactly